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2008 Positivity of operator-matrices of Hua-type
Tsuyoshi Ando
Banach J. Math. Anal. 2(2): 1-8 (2008). DOI: 10.15352/bjma/1240336286


Let $A_j\,\,(j = 1, 2,\ldots , n)$ be strict contractions on a Hilbert space. We study an $n \times n$ operator-matrix: \[\textbf{H}_n(A_1,A_2,\ldots ,A_n) = [(I - A^*_j A_i)^{-1}]^n_{i,j=1}.\] For the case $n = 2$, Hua [Inequalities involving determinants, Acta Math. Sinica, 5 (1955), 463-470 (in Chinese)] proved positivity, i.e., positive semi-definiteness of $\textbf{H}_2(A_1,A_2)$. This is, however, not always true for $n = 3$. First we generalize a known condition which guarantees positivity of $\textbf{H}_n$. Our main result is that positivity of $\textbf{H}_n$ is preserved under the operator M\"obius map of the open unit disc $\mathcal D$ of strict contractions.


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Tsuyoshi Ando. "Positivity of operator-matrices of Hua-type." Banach J. Math. Anal. 2 (2) 1 - 8, 2008.


Published: 2008
First available in Project Euclid: 21 April 2009

zbMATH: 1155.47019
MathSciNet: MR2391242
Digital Object Identifier: 10.15352/bjma/1240336286

Primary: 47B63
Secondary: 15A45 , 47B15

Keywords: Hua theorem , operator-matrix , positivity , strict contraction

Rights: Copyright © 2008 Tusi Mathematical Research Group

Vol.2 • No. 2 • 2008
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